In this paper, we study the existence criteria for ?-bounded solutions of Sylvester matrix dynamical systems on time scales. The advantage of studying this system is it unifies continuous and discrete systems. First, we prove a necessary and sufficient condition for the existence of atleast one ?-bounded solution for Sylvester matrix dynamical systems on time scales, for every Lebesgue ?-deltaintegrable function F, on time scale T+. Further, we obtain a result relating to asymptotic behavior of ?-bounded solutions of this equation. The results are illustrated with suitable examples.

In this study, we establish the complete controllability and observability results for sufficient conditions and necessary conditions for the Sylvester matrix i

This article investigates the exponential stability of the Volterra integro-dynamic (VID) Sylvester matrix system on time scales. The analysis begins by transforming the exponential stability VID Sylvester matrix system into an equivalent Kronecker product VID system using the vectorization operator. Subsequently, the boundedness properties of the solutions are rigorously established. Moreover, the theoretical results formulated on an arbitrary time scale unify and extend the stability conditions for both integral and discrete Volterra equations. Finally, the efficacy and validity of these findings are substantiated through diverse time scale scenarios, accompanied by a graphical comparative analysis.